Set-valued generalizations of Baire's category theorem
The classical Mittag-Leffler theorem asserts that if \(\{X_k\}\) is a sequence of complete metric spaces and \(f_k: X_k\to X_{k-1}\) is a continuous function with \(f_k (X_k)\) dense in \(X_{k-1}\), then \(\bigcap^\infty_{k=1} f_1\circ \dots \circ f_k(X_k)\) is dense in \(X_0\). In this well-written article the author extends the Mittag-Leffler theorem to Čech-complete topological spaces and continuous closed-valued multifunctions with dense ranges. The methods of Stone-Čech compactification of the spaces \(X_i\), \(i=1, 2,\dots\) are incorporated. An example showing the necessity of the closedness of the values of multifunctions is provided. A question is asked whether the continuity may be replaced by lower semicontinuity.
scientific article; zbMATH DE number 861795